# American Institute of Mathematical Sciences

November  2016, 10(4): 797-809. doi: 10.3934/amc.2016041

## Modelling the shrinking generator in terms of linear CA

 1 Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas (UNICAMP), R. Sérgio Buarque de Holanda, 651, Cidade Universitária, Campinas - SP, 13083-859 2 Instituto de Tecnologías Físicas y de la Información, Consejo Superior de Investigaciones Científicas, C/Serrano 144, 28006, Madrid, Spain

Received  December 2014 Revised  June 2016 Published  November 2016

This work analyses the output sequence from a cryptographic non-linear generator, the so-called shrinking generator. This sequence, known as the shrunken sequence, can be built by interleaving a unique PN-sequence whose characteristic polynomial serves as basis for the shrunken sequence's characteristic polynomial. In addition, the shrunken sequence can be also generated from a linear model based on cellular automata. The cellular automata here proposed generate a family of sequences with the same properties, period and characteristic polynomial, as those of the shrunken sequence. Moreover, such sequences appear several times along the cellular automata shifted a fixed number. The use of discrete logarithms allows the computation of such a number. The linearity of these cellular automata can be advantageously employed to launch a cryptanalysis against the shrinking generator and recover its output sequence.
Citation: Sara D. Cardell, Amparo Fúster-Sabater. Modelling the shrinking generator in terms of linear CA. Advances in Mathematics of Communications, 2016, 10 (4) : 797-809. doi: 10.3934/amc.2016041
##### References:
 [1] S. D. Cardell and A. Fúster-Sabater, Cryptanalysing the shrinking generator, Proc. Comp. Sci., 51 (2015), 2893-2897. [2] S. D. Cardell and A. Fúster-Sabater, Performance of the cryptanalysis over the shrinking generator, in Int. Joint Conf. CISIS'15 and ICEUTE'15 (eds. A.H. et al.), Springer, 2015, 111-121. [3] S. D. Cardell and A. Fúster-Sabater, Linear models for the self-shrinking generator based on CA, J. Cell. Autom., 11 (2016), 195-211. [4] K. Cattell and J. C. Muzio, One-dimensional linear hybrid cellular automata, IEEE Trans. Comp.-Aided Des., 15 (1996), 325-335. doi: 10.1109/12.508317. [5] D. Coppersmith, H. Krawczyk and Y. Mansour, The shrinking generator, in Adv. Crypt. - CRYPTO '93, Springer-Verlag, 1993, 23-39. doi: 10.1007/3-540-48329-2_3. [6] A. K. Das, A. Ganguly, A. Dasgupta, S. Bhawmik and P. P. Chaudhuri, Efficient characterisation of cellular automata, IEEE Proc. Comp. Dig. Techn., 137 (1990), 81-87. [7] S. Das and D. RoyChowdhury, Car30: A new scalable stream cipher with rule 30, Crypt. Commun., 5 (2013), 137-162. doi: 10.1007/s12095-012-0079-1. [8] P. F. Duvall and J. C. Mortick, Decimation of periodic sequences, SIAM J. Appl. Math., 21 (1971), 367-372. [9] A. Fúster-Sabater and P. Caballero-Gil, Linear solutions for cryptographic nonlinear sequence generators, Phys. Lett. A, 369 (2007), 432-437. doi: 10.1063/1.2827050. [10] A. Fúster-Sabater, M. E. Pazo-Robles and P. Caballero-Gil, A simple linearization of the self-shrinking generator by means of cellular automata, Neural Netw., 23 (2010), 461-464. [11] S. W. Golomb, Shift Register-Sequences, Aegean Park Press, Laguna Hill, California, 1982. [12] J. Jose, S. Das and D. RoyChowdhury, Inapplicability of fault attacks against trivium on a cellular automata based stream cipher, in 11th Int. Conf. Cell. Autom. Res. Ind. ACRI 2014, Springer-Verlag, 2014, 427-436. [13] A. Kanso, Modified self-shrinking generator, Comp. Electr. Engin., 36 (2010), 993-1001. [14] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, New York, NY, 1986. [15] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory, 15 (1969), 122-127. [16] W. Meier and O. Staffelbach, Analysis of pseudo random sequences generated by cellular automata, in Adv. Crypt. - EUROCRYPTO '91, Springer-Verlag, Berlin, 1991, 186-199. doi: 10.1007/3-540-46416-6_17. [17] W. Meier and O. Staffelbach, The self-shrinking generator, in Adv. Crypt. - EUROCRYPT 1994, Springer-Verlag, 1994, 205-214. doi: 10.1007/BFb0053436. [18] A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, FL, 1996. [19] M. Mihaljević, Y. Zheng and H. Imai, A fast and secure stream cipher based on cellular automata over GF(q), in Proc. Global Telecomm. Conf. GLOBECOM 1998, 1998, 3250-3255. [20] C. Paar and J. Pelzl, Understanding Cryptography, Springer, Berlin, 2010. [21] S. Wolfram, Cellular automata as simple self-organizing system, Caltrech preprint, CALT-68-938, 1982. [22] S. Wolfram, Cryptography with cellular automata, in Adv. Crypt. - EUROCRYPT 1985, Springer-Verlag, 1985, 429-432.

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##### References:
 [1] S. D. Cardell and A. Fúster-Sabater, Cryptanalysing the shrinking generator, Proc. Comp. Sci., 51 (2015), 2893-2897. [2] S. D. Cardell and A. Fúster-Sabater, Performance of the cryptanalysis over the shrinking generator, in Int. Joint Conf. CISIS'15 and ICEUTE'15 (eds. A.H. et al.), Springer, 2015, 111-121. [3] S. D. Cardell and A. Fúster-Sabater, Linear models for the self-shrinking generator based on CA, J. Cell. Autom., 11 (2016), 195-211. [4] K. Cattell and J. C. Muzio, One-dimensional linear hybrid cellular automata, IEEE Trans. Comp.-Aided Des., 15 (1996), 325-335. doi: 10.1109/12.508317. [5] D. Coppersmith, H. Krawczyk and Y. Mansour, The shrinking generator, in Adv. Crypt. - CRYPTO '93, Springer-Verlag, 1993, 23-39. doi: 10.1007/3-540-48329-2_3. [6] A. K. Das, A. Ganguly, A. Dasgupta, S. Bhawmik and P. P. Chaudhuri, Efficient characterisation of cellular automata, IEEE Proc. Comp. Dig. Techn., 137 (1990), 81-87. [7] S. Das and D. RoyChowdhury, Car30: A new scalable stream cipher with rule 30, Crypt. Commun., 5 (2013), 137-162. doi: 10.1007/s12095-012-0079-1. [8] P. F. Duvall and J. C. Mortick, Decimation of periodic sequences, SIAM J. Appl. Math., 21 (1971), 367-372. [9] A. Fúster-Sabater and P. Caballero-Gil, Linear solutions for cryptographic nonlinear sequence generators, Phys. Lett. A, 369 (2007), 432-437. doi: 10.1063/1.2827050. [10] A. Fúster-Sabater, M. E. Pazo-Robles and P. Caballero-Gil, A simple linearization of the self-shrinking generator by means of cellular automata, Neural Netw., 23 (2010), 461-464. [11] S. W. Golomb, Shift Register-Sequences, Aegean Park Press, Laguna Hill, California, 1982. [12] J. Jose, S. Das and D. RoyChowdhury, Inapplicability of fault attacks against trivium on a cellular automata based stream cipher, in 11th Int. Conf. Cell. Autom. Res. Ind. ACRI 2014, Springer-Verlag, 2014, 427-436. [13] A. Kanso, Modified self-shrinking generator, Comp. Electr. Engin., 36 (2010), 993-1001. [14] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, New York, NY, 1986. [15] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory, 15 (1969), 122-127. [16] W. Meier and O. Staffelbach, Analysis of pseudo random sequences generated by cellular automata, in Adv. Crypt. - EUROCRYPTO '91, Springer-Verlag, Berlin, 1991, 186-199. doi: 10.1007/3-540-46416-6_17. [17] W. Meier and O. Staffelbach, The self-shrinking generator, in Adv. Crypt. - EUROCRYPT 1994, Springer-Verlag, 1994, 205-214. doi: 10.1007/BFb0053436. [18] A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, FL, 1996. [19] M. Mihaljević, Y. Zheng and H. Imai, A fast and secure stream cipher based on cellular automata over GF(q), in Proc. Global Telecomm. Conf. GLOBECOM 1998, 1998, 3250-3255. [20] C. Paar and J. Pelzl, Understanding Cryptography, Springer, Berlin, 2010. [21] S. Wolfram, Cellular automata as simple self-organizing system, Caltrech preprint, CALT-68-938, 1982. [22] S. Wolfram, Cryptography with cellular automata, in Adv. Crypt. - EUROCRYPT 1985, Springer-Verlag, 1985, 429-432.
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